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Chapter 4: Problem 9
Let \(X\) be \(\chi^{2}(50)\). Approximate \(P(40
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Let \(X\) be \(N\left(\mu, \sigma^{2}\right)\) and consider the transformation \(X=\log (Y)\) or, equivalently, \(Y=e^{X}\) (a) Find the mean and the variance of \(Y\) by first determining \(E\left(e^{X}\right)\) and \(E\left[\left(e^{X}\right)^{2}\right]\), by using the mgf of \(X\). (b) Find the pdf of \(Y\). This is the pdf of the lognormal distribution.
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of size \(n\) from a distribution that is \(N\left(\mu, \sigma^{2}\right)\), where \(\sigma^{2}>0 .\) Show that the sum \(Z_{n}=\sum_{1}^{n} X_{i}\) does not have a limiting distribution.
Suppose \(\mathbf{X}_{n}\) has a \(N_{p}\left(\boldsymbol{\mu}_{n}, \boldsymbol{\Sigma}_{n}\right)\) distribution. Show that $$ \mathbf{X}_{n} \stackrel{D}{\rightarrow} N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma}) \text { iff } \boldsymbol{\mu}_{n} \rightarrow \boldsymbol{\mu} \text { and } \boldsymbol{\Sigma}_{n} \rightarrow \mathbf{\Sigma} $$
. Let \(\mathbf{X}_{n}\) and \(\mathbf{Y}_{n}\) be \(p\) dimensional random vectors. Show that if $$ \mathbf{X}_{n}-\mathbf{Y}_{n} \stackrel{P}{\rightarrow} \mathbf{0} \text { and } \mathbf{X}_{n} \stackrel{D}{\rightarrow} \mathbf{X} $$ where \(\mathbf{X}\) is a \(p\) dimensional random vector, then \(\mathbf{Y}_{n} \stackrel{D}{\rightarrow} \mathbf{X}\)
Let \(\mu\) and \(\sigma^{2}\) denote the mean and variance of the random variable \(X .\) Let \(Y=c+b X\), where \(b\) and \(c\) are real constants. Show that the mean and the variance of \(Y\) are, respectively, \(c+b \mu\) and \(b^{2} \sigma^{2}\).
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